Bound-state soliton and rogue wave solutions for the sixth-order nonlinear Schr\"{o}dinger equation via inverse scattering transform method

TL;DR

This paper applies inverse scattering transform to derive bound-state soliton, rogue wave, and breather solutions for the sixth-order nonlinear Schrödinger equation, analyzing their dynamics and effects of higher-order terms.

Contribution

It introduces a novel inverse scattering transform approach for the sixth-order nonlinear Schrödinger equation with zero and nonzero boundary conditions, deriving explicit solutions.

Findings

Derived bound-state soliton solutions using Laurent series and residue theorem.

Obtained rogue wave and breather solutions via a matrix Riemann-Hilbert problem and Darboux transformation.

Graphical analysis shows how higher-order terms influence wave dynamics.

Abstract

In this work, inverse scattering transform for the sixth-order nonlinear Schr\"{o}dinger equation with both zero and nonzero boundary conditions at infinity is given, respectively. For the case of zero nonzero boundary conditions, in terms of the Laurent's series and generalization of the residue theorem, the bound-state soliton is derived. For nonzero boundary conditions, using the robust inverse scattering transform, we present a matrix Riemann-Hilbert problem of the sixth-order nonlinear Schr\"{o}dinger equation. Then based on the obtained Riemann-Hilbert problem, the rogue wave and breather wave solutions are derived through a modified Darboux transformation. Besides, according to some appropriate parameters choices, several graphical analyses are provided to discuss the dynamical behaviors of the bound-state soliton and rogue wave solutions, and analyse how the higher-order terms affect the bound-state soliton and rogue wave.